Abstract

The propagation of superluminal waves in dispersive media is
investigated, in particular the refraction at surfaces of discontinuity
in layered dielectrics. The negative mass-square of the tachyonic modes
is manifested in the transmission and reflection coefficients, and the
polarization of the incident waves (TE, TM, or longitudinal) can be
determined from the refraction angles. The conditions for total
internal reflection in terms of polarization and tachyon mass are
derived. Brewster angles can be used to discriminate longitudinal from
transversal incidence. Superluminal transmission through dielectric
boundary layers is studied, and the dependence of the intensity maxima
on the transversal and longitudinal refractive indices of the layer is
analyzed. Estimates of the tachyonic plasma frequency and permittivity
of metals are given. The integral version of the tachyonic Maxwell
equations is stated, boundary conditions at the surfaces of
discontinuity are derived for transversal and longitudinal wave
propagation, and singular surface fields and currents are pointed out.
The spectral maps of three TeV γ-ray sources associated with supernova
remnants, which have recently been obtained with imaging atmospheric
Cherenkov detectors, are fitted with tachyonic cascade spectra. The
transversal and longitudinal polarization components are disentangled
in the spectral maps, and the thermodynamic parameters of the
shock-heated ultra-relativistic electron plasma generating the tachyon
flux are extracted from the cascade fits.

1. Introduction

We investigate the refraction of superluminal wave modes in dispersive
media, at dielectric interfaces and boundary layers. The formalism is
developed in analogy to electromagnetic theory [1], even though
there are substantial differences due to the negative mass-square of
tachyons [2], [3], [4] and [5] and the
occurrence of longitudinally polarized modes [6] and [7]. The tachyon
mass shows in deflection angles, transmission coefficients, and in
longitudinal refraction. The superluminal radiation field is a real
Proca field with negative mass-square [8]. In 3D, the
wave modes can be written in terms of field strengths and inductions,
suggesting a counterpart to Maxwell's equations. The tachyonic Maxwell
equations explicitly depend on the scalar and vector potentials, so
that the gauge invariance is broken. We derive differential and
integral field equations, including the material equations in a
dispersive medium relating the tachyonic field strengths and potentials
to inductions via frequency-dependent permeabilities.

The field equations, the polarization of superluminal modes,
and the transversal and longitudinal Poynting vectors of the tachyon
flux in a dispersive medium are discussed in Section 2. The
transversal and longitudinal refractive indices for tachyonic wave
propagation in dielectrics are introduced, and estimates of the
tachyonic conductivity of metals are given. Refraction properties such
as deflection and reflection angles are determined by boundary
conditions on the tachyon potential, the field strengths, and
inductions at the surface of discontinuity. There are two types of
boundary conditions depending on the polarization of the wave fields,
and singular magnetic field strengths and boundary currents can emerge.
This is explained in Section 3.

In Section 4, we
study superluminal refraction at a plane interface generated by a
discontinuity in the permeabilities. The reflection and refraction
angles depend on the polarization of the incident modes, and so do the
transmission rates. The Brewster angles for transversal and
longitudinal incidence are derived, as well as the conditions for total
internal reflection of polarized superluminal radiation. The tachyon
flux through a dielectric boundary layer is investigated, in particular
the intensity peaks at normal incidence. We calculate the transmission
and reflection coefficients, and discuss their frequency dependence and
the effect of polarization.

In Section 5, we
point out evidence for superluminal γ-rays in the spectral maps of
three Galactic TeV sources obtained with the imaging air Cherenkov
detectors HESS and MAGIC [9], [10] and [11]. The
spectra are fitted with nonthermal tachyonic cascades generated by the
shocked electron plasma of the remnants. The transversal and
longitudinal polarization components of the cascades are resolved,
exhibiting extended spectral plateaus in the high GeV region typical
for tachyonic γ-ray emitters, followed by nonthermal power–law slopes
at low TeV energies. The spectral break terminating the GeV plateau is
compared to the break energies of the cosmic-ray spectrum. In Section 6, we
present our conclusions.

2. Tachyonic Maxwell equations and integral
field equations in a permeable medium

The tachyonic radiation field in vacuum is a real vector field
with negative mass-square, satisfying the Proca equation ,
subject to the Lorentz condition .
mt is the mass of the
superluminal Proca field Aμ,
and q the tachyonic charge carried by the
subluminal electron current jμ=(ρ,j).
The mass term is added with a positive sign, and the sign convention
for the metric defining the d’Alembertian ∂ν∂ν
is diag (−1,1,1,1), so that
is the negative mass-square of the radiation field [6] and [12]. The above
wave equation in conjunction with the Lorentz condition is equivalent
to the tachyonic Maxwell equations

(2.1)

where the field strengths are related to the potentials by E=A0-∂A/∂t
and .
In contrast to electromagnetic theory, the Lorentz condition
follows from the field equations and current conservation, ,
cf. after Eq. (2.3).

In a permeable medium, the potentials and field strengths in
the inhomogeneous vacuum equations (2.1) are
replaced by inductions, (A0,A)→(C0,C),
(E,B)→(D,H),
defined by material equations [13] and [14]. We will
mostly consider monochromatic waves, ,
and analogously for the scalar potential, current, charge density,
field strengths, and inductions. Fourier amplitudes are denoted by a
hat. Maxwell's equations in Fourier space read

(2.2)

supplemented by material equations

(2.3)

The inductive potentials
as well as the inductions
and
are related to the primary fields by frequency-dependent dielectric (ε0,ε)
and magnetic (μ0,μ)
permeabilities. The Fourier amplitudes of the field strengths and
potentials are connected by
and .
We take the divergence of the third equation in Eq. (2.2), and
substitute the fourth, to obtain

(2.4)

Current conservation, ,
implies the Lorentz condition ,
or equivalently,
in terms of the inductive potentials.

The sign conventions for the coupling of an electron to the
Proca field are L=-m/γ+q(A0+A·v)
and md(γv)/dt=q(E+v×B),
where γ=(1-υ2)-1/2
is the electronic Lorentz factor and q the
tachyonic charge carried by the electron, cf. after Eq. (2.14). The
primary fields rather than the inductions define the Lorentz force. The
potentials are unambiguously determined by the field strengths and the
dielectric and magnetic permeabilities. We define polarizations ,
,
and ,
relating the inductions and field strengths additively as
and ,
and analogously the potentials
and .
Accordingly,
and ,
with electric susceptibilities κ=ε-1
and κ0=ε0-1.
Analogously,
and ,
with magnetic susceptibilities χ=μ-1
and χ0=μ0-1.
The permeabilities ε0 and μ0
define the inductive potentials in Eq. (2.3), and are
not to be confused with vacuum permeabilities; we use the
Heaviside–Lorentz system, so that ε=ε0=1
and μ=μ0=1
in vacuum.

Applying the Gauss theorem to the divergence equations in Eq. (2.2), as well
as to the Lorentz condition and the potential definition of ,
cf. after Eq. (2.3), we find
the integral field equations

(2.5)

Here, dS=ndS
is the surface element of a closed surface S, the
boundary of a domain V with volume element dV,
and n is the surface normal pointing into
the interior of the domain of integration V.
Current conservation gives

(2.6)

Applying the Stokes theorem to the rotor equations in Eq. (2.2) and the
potential definition of ,
we obtain

(2.7)

Here, dS=ndS
is the surface element of an oriented open surface SL
bounded by the closed loop L, and ds=n×nL ds
is the oriented tangent element of the loop. nL
is the unit vector tangent to the surface and orthogonal to the loop,
pointing to the exterior. n is the unit
normal of the surface at the loop, with the same orientation as the
surface element dS.

To find the transversal and longitudinal dispersion relations,
we use a plane–wave ansatz in the Maxwell equations (2.2) with
vanishing charge and current densities: ,
and analogously for the scalar potential and the field strengths. Here,
k=k(ω)k0,
where k is the wave number to be determined from
the field equations, and k0
is a constant unit vector. k(ω)
as well as k0 can
be complex. The transversality condition is ,
and the set of transversal modes is ,
with amplitude ,
and similarly for the field strengths and the scalar potential. The
dispersion relation determining the transversal wave number is

(2.8)

and the amplitudes of the transversal field strengths follow from :

(2.9)

If the product
does not vanish, the modes must be longitudinal, ,
with dispersion relation

(2.10)

so that .
The amplitudes of the longitudinal scalar potential and the field
strengths read

(2.11)

The amplitudes AT,L(k)
only need to satisfy the transversality/longitudinality condition, that
is, the first equation in Eqs. (2.9) and (2.11),
respectively.

where
stands for .
As above, the superscripts T and L refer to the transversal and
longitudinal flux components. The frequency-dependent permeabilities of
a dispersive transparent medium are real in the absence of energy
dissipation. The flux vectors (2.12) apply
even for negative permeabilities [15], [16] and [17], provided
that we restrict to a frequency range where the squared wave numbers (2.8) and (2.10) are
positive. They are obtained by substituting the polarized plane waves (2.9) and (2.11) into the
Poynting vector ,
and by performing a time average [18].

The transversal and longitudinal phase velocities read ,
and the tachyonic group velocities are ,
so that
and .
In the case of vacuum permeabilities, ε(0)=μ(0)=1,
the transversal and longitudinal velocities coincide, and we find ω=mtγt,
where
denotes the tachyonic Lorentz factor and υgr>1
is the superluminal group velocity. The refractive index nT,L=kT,L/ω,
defined as dimensionless inverse phase velocity, differs for
transversal and longitudinal modes:

(2.13)

At high frequencies ωmt
(with [6] and [18] and =c=1),
we can approximate .
The longitudinal index, ,
has no electromagnetic analog. In the low-frequency regime ωmt,
the refractive indices are frequency dependent even if the
permeabilities stay constant,
and .

To estimate the dielectric permeability for tachyonic wave
propagation, we start with a monochromatic superluminal mode, ,
in a dispersive and possibly dissipative medium. This mode generates
the current ,
,
where σ(ω) is the tachyonic
conductivity of the medium, cf. Eq. (2.14). The
charge distribution
is found by means of current conservation, cf. after Eq. (2.4). We
substitute this current into the field equations (2.2) with
vacuum permeabilities ε(0)=μ(0)=1,
and absorb current and charge density by introducing the permittivity εσ(ω)=1+iσ(ω)/ω.
In this way, we can write the inhomogeneous field equations (2.2) as
and ;
the homogeneous Maxwell equations remain unchanged. The polarization
vector is ,
cf. after Eq. (2.4), and the
London equation
applies. The dispersion relations for the transversal and longitudinal
modes read as in Eqs. (2.8) and (2.10), with ε
replaced by εσ
and ε0=μ(0)=1.

We consider a tachyonic conductivity

(2.14)

where
is the tachyonic plasma frequency, and ne
the electron density of the medium. This is based on Drude's damped
oscillator model, ,
with ,
and E as above [14]. ω0
is the characteristic binding frequency of the electronic oscillators, q
the tachyonic charge, and γ0
is the damping constant related to the tachyonic resistivity ρt
by .
We solve this equation in dipole approximation, neglecting the spatial
dependence of the Fourier components ,
so that σ(ω) in Eq. (2.14) follows
from ,
,
and .
We note ,
where
is the electromagnetic plasma frequency, and αq/αe≈1.4×10-11
the ratio of tachyonic and electric fine structure constants. In the
Heaviside–Lorentz system, αe=e2/(4πc)≈1/137
and αq=q2/(4πc)≈1.0×10-13.
The tachyon–electron mass ratio is mt/m≈1/238;
αq and mt
are estimated from hydrogenic Lamb shifts [6]. We may thus
conclude that the tachyonic conductivity
is by a factor of 10−11 smaller than the
electric counterpart. However, εσ(ω)-1
can still be of order one.

To demonstrate this, we consider a free electron gas, ω0=γ0=0,
so that the tachyonic conductivity (2.14)
simplifies to ,
and the induced permittivity is .
We may write the electromagnetic plasma frequency as ,
where e is the reduced
electronic Compton wavelength, αe
the electric fine structure constant, and ne
the electron density as above. We thus find ,
and the tachyonic counterpart ωp≈3.74×10-6ωp,em.
Metallic electron densities of 1022−1023 cm−3
result in a tachyonic plasma frequency ωp
in the 10−5 eV range. Thus, even though
the tachyonic conductivity is much smaller than the electromagnetic one
at the same frequency, the tachyonic permittivity εσ(ω)
becomes noticeably different from 1 for frequencies comparable to ωp.
Finally, small frequencies of the order of 10−5 eV
do not imply large wavelengths. The tachyonic wavelength is λT,L=2π/kT,L,
with wave numbers defined in Eqs. (2.8) and (2.10). The
maximal transversal/longitudinal wavelength in the medium is thus
and ,
attained in the zero frequency limit, where the permeabilities are of
order one and
is the tachyonic Compton wavelength.

3. Superluminal wave fields at the surface
of discontinuity of dielectrics and conductors

To study tachyon refraction at the interface of two media with
different permeabilities, we have to specify the boundary conditions.
To this end, we split the wave fields into regular and singular parts,
writing the scalar induction and the magnetic field strength as

(3.1)

The same decomposition is used for the scalar and vector potentials ,
the electric field strength ,
and the vectorial inductions ,
,
and .
Here, θ(S) is the Heaviside step
function, and S(x)=0
is the surface separating the media defined by permeabilities (ε0,1,ε1,μ0,1,μ1)
and (ε0,2,ε2,μ0,2,μ2),
cf. Eq. (2.3), which can
be frequency dependent (dispersive) and complex (dissipative). The
subscripts 1 and 2 on the fields and permeabilities refer to the
respective media; in the decomposition (3.1) of the
regular field components, it is understood that domain S(x)>0
is occupied by medium 2, so that
for S>0, and
applies in domain S<0 where medium 1 is
located. The normal vector, n:=S/|S|, thus points into
the interior of medium 2. The singular part of the field, if any, is a
distribution supported on the boundary surface S=0,
typically containing a Dirac function δ(S)
as factor. In contrast to the regular components in Eq. (3.1), the
singular field strengths and inductions do not show in the material
equations (2.3). In
Sections 3.1 and 3.2, we assume
vanishing charge and current densities in the field equations, as well
as the absence of a singular surface current at the interface. In Section 3.3, we
consider inhomogeneous boundary conditions consistent with currents.

3.1. Boundary conditions for transversal
tachyons

We start by substituting ansatz (3.1) into the
field equations (2.2) (with zero
current and charge density) and the Lorentz condition, and assume that
the fields subjected to rotors and divergences have no singular part.
That is, we try to solve under the condition that ,
,
,
,
and
are regular at the interface, obtaining five relations to be satisfied
at the boundary S=0,

Assuming the potentials
and
to be non-singular, we find, from the potential representation of
and ,
cf. after Eq. (2.3),

(3.6)

All fields are regular at the boundary surface, with exception of ,
whose singular part is obtained from (3.2) or (3.4).
does not vanish except for a special polarization (TE waves), but no
singular contribution can occur in the transversal flux vector (2.12), as
is identically zero in both media, cf. Eq. (2.9). The
boundary conditions on transversal modes are defined by Eqs. (3.3), (3.5) and (3.6).

We proceed analogously to the transversal case, but now
assuming all fields to be regular at the boundary except for the
magnetic field .
On substituting ansatz (3.1) into the
field equations (2.3) and (2.4) and the
Lorentz condition, cf. after Eq. (2.4), we find

where we used (3.9). Eqs. (3.7), (3.10), (3.11) and (3.12)
constitute the boundary conditions for longitudinal modes. The regular
longitudinal
field vanishes identically in both media, cf. Eq. (2.11), but a
singular surface field
emerges, which, however, does not affect the longitudinal flux vector (2.12), since
all other field strengths, potentials, and inductions are regular.

The transversal and longitudinal boundary conditions derived
in Sections 3.1 and 3.2 apply at the
interface of two media with different permeabilities discontinuous at
the interface. They unambiguously determine the refractive properties
of tachyons, such as deflection and reflection angles, and assure
transmission and reflection ratios consistent with energy conservation
in non-absorptive media, cf. Section 4.

3.3. Singular boundary currents and charge
densities

In the case of non-vanishing charge and current densities in
field equations (2.2), the
transversal and longitudinal boundary conditions derived in Sections 3.1 and 3.2 become
inhomogeneous due to singular surface currents. As in (3.1), we split
current and charge density into a regular and singular part,

(3.13)

The singular surface charge and current,
and ,
are supported at the interface S=0. The subscripts
1 and 2 refer to the respective medium as defined after Eq. (3.1). We
substitute ansatz (3.13) into the
continuity equation, cf. after Eq. (2.4), to find
the boundary condition required by current conservation,

(3.14)

Three of the transversal boundary conditions in Section 3.1 have
to be modified in the presence of currents. Condition (3.2) is
replaced by

As for the longitudinal boundary conditions in Section 3.2,
there are two changes. The first condition in Eq. (3.7) is
replaced by Eq. (3.15) with ,
and the third condition in Eq. (3.7) by Eq. (3.16). It is
easy to check that boundary condition (3.16) is
consistent with current conservation (3.14).

4. Tachyon refraction at plane interfaces
of dispersive media

We take the z=0 plane as the interface S
separating medium 2 in the upper half-space z>0
from medium 1 in the lower half-space. The media are defined by
frequency-dependent permeabilities (ε0,1,ε1,μ0,1,μ1)
and (ε0,2,ε2,μ0,2,μ2),
respectively, cf. Eq. (2.3) and after
Eq. (3.1). We
consider a tachyonic plane wave, cf. after Eq. (2.7), incident
from the lower half-space upon the interface, the e1,2
plane. The wave number kin
of this incoming transversal or longitudinal wave is defined by
dispersion relation (2.8) or (2.10), with
permeabilities carrying subscript 1. Part of the wave is reflected back
into medium 1, and the wave number kre
of the reflected wave coincides with kin.
The wave number ktr of the
wave transmitted into the upper half-space is determined by the
permeabilities of medium 2. The wave vectors of the respective modes
are denoted by kin=kink0,in,
kre=krek0,re,
and ktr=ktrk0,tr,
where the zero subscript indicates unit vectors. We assume the incoming
plane wave to be homogeneous, so that k0,in
is a real unit vector; the wave numbers kin,tr
can be complex. Since k0,in
is real, the unit wave vector k0,re
of the reflected wave is real too, as shown below. The transmitted wave
is in general inhomogeneous if the wave number in medium 1 or 2 is
complex, so that k0,tr
is a complex unit vector, .
We adopt the convention Re(kin,tr)>0,
since wave numbers are only defined as squares by the dispersion
relations. If medium 1 in the lower half-space has real permeabilities,
the vacuum for instance, then the incident wave number kin
is real.

We choose the incoming real unit wave vector k0,in
in the e1,3 plane.
(The ei
are coordinate unit vectors.) The normal vector of the interface is e3,
pointing into medium 2. A convenient angular parametrization of the
wave vectors is

(4.1)

The incoming wave moves through the lower half-space towards the
interface at z=0, so that
and
are positive, kin=kre,
,
and .
The angles
and
in Eq. (4.1) are in
general complex, and θre=π-θin.
The boundary conditions at z=0 can only be
satisfied if the phase factors eik·x
of the three waves coincide at the boundary. This requires kin·e1=kre·e1=ktr·e1,
and the same for e2
at the interface. (The wave vectors are orthogonal to e2,
as k0,in is by
definition.) Hence, ,
which is the tachyonic counterpart to Snell's reflection law [19] and [20].

The refractive indices of medium 1 and 2 are denoted by n1,2=kin,tr/ω,
cf. Eq. (2.13), and
their ratio by .
This applies to transversal as well as longitudinal indices, e.g., .
The transversal/longitudinal refraction law can thus be written as .
In the high-frequency regime ωmt,
the transversal refractive index ratio simplifies to ,
and the longitudinal one to ,
cf. after Eq. (2.13). In the
low-frequency limit, ωmt,
the refraction angle θtr is
determined by
or .
If we consider dielectrics with μ(0)=ε0=1,
the longitudinal refractive index ratio
at low frequencies is just the inverse of
at high frequency. The refraction law can thus be used to discriminate
between transversal and longitudinal polarization.

4.1. Refraction of superluminal TE and TM
waves

We first consider tachyonic TE waves, so that the incoming
mode
is linearly polarized, with amplitude orthogonal to the plane of
incidence generated by the normal vector e3
of the boundary and the wave vector k0,in
in the e1,3 plane.
Thus, ,
cf. Eq. (2.9). The
boundary conditions stated in Section 3.1 are
satisfied by the reflected and transmitted waves, which are likewise
polarized in the e2
direction, so that the respective modes are Eree2eikre·x
and Etre2eiktr·x.
The wave numbers of the incident and reflected waves are defined by the
transversal dispersion relation (2.8) of medium
1 with permeabilities (ε0,1,ε1,μ0,1,μ1),
cf. Eq. (2.3). The wave
number of the transmitted wave is calculated with the permeabilities (ε0,2,ε2,μ0,2,μ2)
of medium 2 in the upper half-space. The boundary conditions (3.3), (3.5) and (3.6) give, if
combined with Snell's law as stated above, two independent relations
for the three amplitudes:

(4.2)

obtained via
and .
We write this in amplitude ratios by means of Eq. (4.1):

(4.3)

Using Snell's law, we parametrize by the incidence angle, substituting .
Even though we employ here and in the following only two boundary
conditions, the remaining ones are satisfied as well, by virtue of the
above refraction laws.

We turn to superluminal TM waves, where the electric field is
linearly polarized parallel to the plane of incidence. It is convenient
to write the boundary conditions in terms of the magnetic field, by way
of ,
where
is orthogonal to the plane of incidence, cf. Eq. (2.9).
Accordingly, ,
and analogously for the reflected and transmitted modes. The boundary
conditions for transversal modes again give two independent relations
among the three amplitudes,

(4.4)

Here, we used the same two boundary conditions as for TE modes. The
amplitude ratios read

(4.5)

which differ from the TE ratios just by an interchange of μ1ktr
and μ2kin,
and we substitute
stated after Eq. (4.3) to
parametrize by the incidence angle.

The energy flux (i.e., the incident, reflected, or transmitted
energy per unit time and unit surface area) carried by a superluminal
mode with real unit wave vector k0
is FT,L:=|ST,L||k0·n|.
The transversal and longitudinal flux vectors ST,L
are defined in Eq. (2.12), with the
respective incident, reflected, or transmitted wave substituted. As for
the transmitted wave, we assume the two media to be non-dissipative, so
that k0,tr
is real. The superscripts T and L denote transversal and longitudinal
waves, the latter are studied in Section 4.2. The
transversal and longitudinal reflection and transmission coefficients
are defined by the flux ratios

(4.6)

where the subscripts indicate the respective fields (incident,
reflected, or transmitted) to be substituted into the flux vector.
Energy conservation requires RT,L+TT,L=1.
(We do not define a transmission coefficient for dissipative media.) By
making use of the transversal flux vector (2.12), we find
the reflection and transmission ratios of tachyonic TE and TM modes as

(4.7)

with the amplitude ratios (4.3) and (4.5)
substituted. It is easy to check that energy is conserved, which
suggests that we have got the boundary conditions in Section 3.1
right.

where
is the ratio of the transversal refractive indices of medium 1 and 2,
and ,
cf. after Eq. (4.1). In the
high-frequency regime ωmt,
we approximate ,
cf. after Eq. (2.13), and find
the reflection coefficients for TE and TM waves as

In this limit, the reflected fraction of the transversal tachyon flux
is determined by the magnetic permeabilities only. Transversal
refraction will further be discussed after Eq. (4.16), together
with the longitudinal reflection coefficients derived in Section 4.2.

We start with a longitudinal incident mode, ,
and use analogous notation for the reflected and transmitted fields.
All wave vectors lie in the e1,3
plane. The permeabilities of media 1 and 2 are labeled as indicated
before Eq. (4.2). The
boundary conditions (3.7), (3.10), (3.11) and (3.12) give two
independent relations for the amplitudes,

(4.11)

derived from
and .
The amplitude ratios read accordingly

(4.12)

with
defined after Eq. (4.3). The
singular surface magnetic field
is calculated via Eq. (3.9):

(4.13)

The longitudinal reflection and transmission coefficients defined in
Eqs. (2.12) and (4.6) are

(4.14)

where we substitute the ratios (4.12). The
transmission coefficient applies for real permeabilities, energy being
conserved in non-absorptive media, RL+TL=1.

where ,
,
and
is the quotient nL,2/nL,1
of the longitudinal refractive indices, cf. Eq. (2.13). At high
frequency ωmt,
we find ,
and for ωmt
the refractive index becomes ,
so that the longitudinal reflection coefficients read in the respective
limit

(4.16)

If we consider vacuum permeabilities in medium 1, ε(0),1=μ(0),1=1,
and a dielectric permeability ε2
different from one in medium 2 (with ε0,2=μ(0),2=1),
we find
and a finite reflectivity, ,
for longitudinal modes in the low-frequency regime. The same finite
reflectivity applies for tachyonic TE and TM modes, but in the opposite
limit, ωmt,
with different ε2(ω),
cf. Eq. (4.9). The
transversal reflection coefficients Eq. (4.10) valid for
ωmt
vanish, as the indicated leading order of the frequency expansion is
independent of ε2.

We consider two other special cases. First, the case where the
refracted wave vector is orthogonal to the reflected wave, θre-θtr=π/2,
so that θtr+θin=π/2,
and thus
and .
This Brewster incidence angle, ,
follows from the refraction law stated after Eq. (4.1);
denotes the transversal or longitudinal refractive index ratio, cf.
after Eqs. (4.8) and (4.15). The
refractive indices are assumed to be real in both media. As for TM
waves, we find Hre=0
in Eq. (4.5), provided
that the magnetic permeabilities μ1
and μ2 of the two media
coincide. Similarly for longitudinally polarized waves, Ere=0
in Eq. (4.12), provided
that ε0,1=ε0,2.
At this incidence angle, the energy of a tachyonic TM wave or a
longitudinal wave is fully transmitted. If the incident transversal
wave is elliptically polarized (being a complex linear combination of
TE and TM waves), the reflected wave is a TE wave linearly polarized
orthogonal to the plane of incidence. If we do not require μ1=μ2,
and define the incidence angle by Hre=0,
we find

(4.17)

The longitudinal incidence angle defined by Ere=0
is likewise given by Eq. (4.17), with
replaced by ,
and
by ,
cf. Eqs. (4.8) and (4.15).

The second special case is total internal reflection, which
requires real wave numbers and incidence angles satisfying ,
so that
defined after Eq. (4.3) is zero or
imaginary with
to ensure damping. (More generally, the damping condition for a
transmitted wave in medium 2 is .)
Thus k0,tr is a
complex unit vector, even though the permeabilities in both media are
real; its e1
component
is found via Snell's law, cf. after Eq. (4.1). The
reflection coefficients RT,L
in Eqs. (4.7) and (4.14) are equal
to 1, so that the incident flux is totally reflected. The refracted
wave in medium 2 is exponentially damped along the z
axis, and no energy is transmitted. Internal reflection can only occur
if ,
that is, medium 1 must be optically thicker than medium 2 for
transversal or longitudinal modes. A third special case, normal
incidence on a boundary layer of finite thickness separating two
dielectric media, is discussed in the next subsection.

4.3. Normal incidence: reflection and
transmission of tachyons at a boundary layer

We consider three dispersive media separated by parallel
boundary planes z=0 and z=h.
Medium 1 lies in the lower half-space, medium 2 is a layer of thickness
h located in 0<z<h,
and medium 3 fills the half-space z>h.
The respective permeabilities and refractive indices are denoted by
subscripts, ε1,2,3,
etc., cf. after Eq. (3.1). The layer
is hit by a tachyonic plane wave propagating in the lower half-space
orthogonally incident upon the z=0 plane. To
satisfy the boundary conditions at the two interfaces, we start with
the ansatz

(4.18)

where the Ei are fields in
the respective media. The notation is explained at the beginning of Section 4 and in
the previous two subsections. The transversal field strengths
are complex linear combinations of the linearly polarized fields Eie1
and Eie2,
and the longitudinal ones read Eie3.
Thus, E1ek
is a superposition of the incoming and reflected waves in medium 1, the
wave number in this medium being kin.
Similarly, E2ek
is composed of the wave transmitted into medium 2 and a second wave
arising by reflection at the second interface z=h.
Finally, E3ek
is the outgoing wave in medium 3. ktr
and kout are the wave
numbers in medium 2 and 3, respectively. All wave numbers have a
positive real part, so that the negative sign in the exponents of the
reflected waves implies the unit wave vector −e3.
We take the amplitude of the incident wave Ein
as input parameter; the remaining four amplitudes are obtained from the
boundary conditions at the two interfaces.

4.3.1. Transversally polarized superluminal
modes

On substituting αT into the
first and second equations in Eq. (4.19), we find

(4.21)

Reflection and transmission coefficients are defined as in Eqs. (4.6) and (4.7),
and ,
so that the transversal ratios read

(4.22)

As for the transmission coefficient, we assume real permeabilities and
wave numbers in media 1 and 3, so that damping can only occur in the
boundary layer, that is medium 2. If all three media are dielectrics,
energy conservation applies, RT+TT=1.
In this case, the coefficients are periodic in h,
the thickness of the boundary layer, and the extrema of RT
and TT are determined by sin(2hktr(ω))=0,
the variation being with respect to the layer thickness, ∂RT/∂h=0.
Thus the intensity minima and maxima occur at frequencies where the
transversal wave number ktr=ωnT,2
is an integer multiple of π/(2h);
nT,2(ω)
is the transversal refractive index (2.13) of the
layer. We find, for cos(2hktr)=±1,

(4.23)

where μi denotes the
magnetic permeability and nT,i
the transversal refractive index of the respective medium, cf. after
Eq. (2.13). In the
high-frequency limit ωmt,
we substitute
in Eq. (4.23),

If all permeabilities are real, energy is conserved, RL+TL=1,
and these coefficients are periodic in the layer thickness h.
The intensity extrema of RL
and TL are defined by the
longitudinal wave number ktr=ωnL,2(ω)
in the layer, cf. Eqs. (2.10) and (2.13),
occurring at frequencies solving cos(2hktr(ω))=±1,
analogously to Eq. (4.23):

(4.30)

where nL,i
is the longitudinal refractive index of the respective medium. At high
frequencies, ωmt,
we substitute
to find the extremal reflection coefficients for longitudinal tachyons,

(4.31)

In the low-frequency regime, we approximate
so that Eq. (4.30)
simplifies to

(4.32)

For example, we may set all permeabilities equal to one apart from the
permittivity ε2 of the
layer. At high frequency, the longitudinal flux is almost totally
transmitted since the leading order of the reflection coefficient
vanishes, .
At low frequency, we still have
(that is, for wave numbers ktr=(l+1/2)π/h
with integer l), but there is a non-vanishing
fraction
of the incident flux reflected at frequencies satisfying ktr(ω)=lπ/h.
This is just the opposite of the transversal case in Eqs. (4.24) and (4.25), where
at high frequencies, whereas
for ωmt.
More generally, there is a symmetry in the reflection coefficients (4.23) and (4.30) with
regard to the interchange ε0↔μ,
ε↔μ0,
which is apparent in the asymptotic limits (4.24) and (4.31) as well
as Eqs. (4.25) and (4.32). However,
the extremal frequencies defined by the zeros of sin(2hktr(ω))
differ for transversal and longitudinal modes, unless the wave numbers (2.8) and (2.10) coincide
in the layer.

where d is the distance to the source, and pT,L(ω=E/)
the transversal/longitudinal tachyonic spectral density averaged over a
nonthermal electronic power-law distribution, [21]. As for the
latter, α is the electronic power-law index, the
ultra-relativistic electronic Lorentz factors range in an interval γ1γ<∞,
γ11,
and the exponential cutoff is related to the electron temperature by β=mc2/(kT).
A thermal Maxwell–Boltzmann distribution corresponds to α=−2
and γ1=1. The least-squares
fit is performed with the unpolarized flux density dNT+L=dNT+dNL,
and then split into transversal and longitudinal radiation components.
The details of the spectral fitting have been explained in Ref. [22]. The
parameters of the electron distributions dρα,β
generating the tachyonic cascades are listed in Table 1. The
cutoff parameter β in the Boltzmann factor could
not be extracted from the presently available flux points, in contrast
to the power–law index α and the lower edge γ1
of Lorentz factors. The power–law slope ultimately terminates in
exponential decay, cf. the spectral map of HESS J1825−137 in Fig. 5 of
Ref. [7]. As for the
spectral fits in Fig. 1, Fig. 2 and Fig. 3, there is
no downward bend yet in the presently accessible TeV range.

Fig. 1. Spectral map of the TeV γ-ray
source HESS J1837−069 associated with the pulsar wind nebula AX
J1838.0−0655. Flux points from Ref. [9]. The solid
line depicts the unpolarized differential tachyon flux dNT+L/dE
rescaled with E2 tachyon
flux dNT+L/dE
rescaled with E2, cf. (5.1).
The transversal (dot-dashed) and longitudinal (double-dot-dashed) flux
densities dNT,L/dE
add up to the total unpolarized flux cascade ρ1=T+L
generated by a nonthermal electron population. The cascade admits a
power–law slope ∝E1-α
with electron index α≈1.4. A spectral break at
is visible as edge in the longitudinal component, where
is the tachyon mass [6] and [18]. The
least-squares fit is based on the unpolarized tachyon flux T+L, cf. Table 1.

Fig. 2. Spectral map of the extended TeV source HESS
J1834−087 in supernova remnant W41. HESS data points from Ref. [9], MAGIC
points from Ref. [10]. Notation
as in Fig. 1. The
parameters of the shock-heated electron plasma are listed in Table 1. The
spectral break in the longitudinal flux component (L) of the cascade
occurs at 0.80 TeV. The distance estimate of this source is
4 kpc, and its electron index is 1.9, quite similar to the TeV
source in Fig. 3 at a
comparable distance. The power-law slope is steeper than of HESS
J1837−069 at 6.6 kpc, cf. Fig. 1; there is
no interstellar absorption of the tachyon flux [22].

α is the electron index, and γ1
the lower threshold Lorentz factor of the ultra-relativistic electron
populations, cf. after Eq. (5.1).
determines the amplitude of the tachyon flux, from which the electron
count ne is inferred at the
indicated distance d, cf. Refs. [[27], [28] and [29]]. The
parameters α, γ1,
and
are extracted from the χ2-fit
T+L in the figures. The amplitude
is related to the electron number by ,
cf. Ref. [7].

Fig. 1 shows the
TeV spectral map of the unidentified TeV source HESS J1837−069 [9], coincident
with the pulsar wind nebula AX J1838.0−0655. The distance estimate to
this X-ray nebula is 6.6 kpc, by association with a nearby
cluster of red supergiants. Fig. 2 depicts
the spectral fit to the extended TeV source HESS J1834−087, associated
with the shell-type supernova remnant W41, cf. Refs. [9] and [10]. The
kinematic distance estimate of W41 is 4 kpc. Spectral plateaus
in the MeV to GeV range occur frequently in spectral maps of both
thermal and nonthermal TeV sources, and can easily be fitted with
tachyonic cascade spectra, in contrast to electromagnetic or hadronic
radiation models. Thermal spectra of γ-ray binaries such as binary
pulsars and microquasars are studied in Refs. [12] and [23], and a
thermal cascade fit of a γ-ray quasar is performed in Ref. [24]. The
shocked electron plasma of supernova remnants requires nonthermal
electron densities. Fig. 3 shows a
nonthermal cascade fit to the TeV source HESS J1813−178, located in the
vicinity of the H II region W33 [9] and [11] at a
distance of 4.5 kpc. The lower edge of Lorentz factors of the
electron plasma is γ1≈3.0×109,
inferred from the cascade fit. The corresponding electron and proton
energies are
and .
These lower bounds on the energy of the radiating source particles are
to be compared to the spectral breaks in the cosmic-ray spectrum at 1015.5
and 1017.8 eV, dubbed knee and second
knee, respectively [25] and [26]. The bounds
are one order lower for the sources in Fig. 1 and Fig. 2, cf. Table 1, but in
all three remnants the lower bound on the proton energy is close to the
second knee.

6. Conclusion

We have studied the refraction of superluminal radiation at
dielectric interfaces, in particular the refraction angles for
transversal and longitudinal incidence, and the dependence of the
transmission and reflection coefficients on the polarization of the
incident radiation modes. Speed and energy of tachyonic quanta are
related by
in vacuum, cf. after Eq. (2.12). At γ-ray
energies, their speed is close to the speed of light, the basic
difference to electromagnetic radiation being the longitudinally
polarized flux component. The polarization of tachyons can be
determined from the refraction angles at dielectric interfaces, cf.
after Eq. (4.1), or from
the reflection coefficients, which greatly differ for transversal and
longitudinal modes, cf. after Eq. (4.32). We
performed tachyonic cascade fits to the γ-ray
spectra of the TeV sources in Fig. 1, Fig. 2 and Fig. 3, and
disentangled the transversal and longitudinal flux components.

Shocked electron plasmas generate nonthermal γ-ray
cascades typical for supernova remnants and pulsar wind nebulae. The
characteristic feature is the extended spectral plateau at GeV
energies, followed by a steep but barely bent spectral slope in the low
TeV range, assuming a double-logarithmic and E2-rescaled
flux representation as in Fig. 1, Fig. 2 and Fig. 3. The
spectral maps discussed here are to be compared to the unpulsed γ-ray
spectrum of the Crab Nebula, cf. Fig. 1 in Ref. [18], the
spectral map of supernova remnant RX J1713.7−3946 in Fig. 2 of Ref. [18], the
spectra of HESS J1825−137 and TeV J2032+4130 in Figs. 5 and 6 of Ref. [7], and the
extended γ-ray cascade of supernova remnant W28 in Fig. 4 of Ref. [22]. All these
spectra show GeV plateaus followed by straight or slightly curved
power–law slopes. Traditional radiation mechanisms such as inverse
Compton scattering or proton–proton scattering followed by pion decay
fail to reproduce the extended plateaus in the spectral maps, a fact
often concealed by compression in broadband maps. By contrast,
tachyonic cascades provide excellent fits to the GeV plateaus and
power–law slopes. The latter are a signature of shock-heated electron
plasmas, and absent in the spectra of thermal γ-ray sources like TeV
blazars [30] and [31], where the
plateaus terminate in exponential decay without power–law transition.
The spectral breaks at the join of the spectral plateaus and the
power–law slopes are determined by the lower threshold Lorentz factors
of the nonthermal source populations in the remnants. These Lorentz
factors can be extracted from the spectral fits [32], and
suggest that TeV γ-ray sources in Galactic supernova remnants are
capable of accelerating protons to energies above the spectral break at
1017.8 eV in the cosmic-ray spectrum.

Acknowledgments

The author acknowledges the support of the Japan Society for
the Promotion of Science. The hospitality and stimulating atmosphere of
the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy,
and the Institute of Mathematical Sciences, Chennai, are likewise
gratefully acknowledged. I also thank the referee for useful
suggestions regarding substance as well as readability, which greatly
helped to improve the initial draft.